Method and system for operating an atomic clock with reduced spin-exchange broadening of atomic clock resonances

ABSTRACT

The present invention relates to a method and system for using end resonances of highly spin-polarized alkali metal vapors for an atomic clock, magnetometer or other system. A left end resonance involves a transition from the quantum state of minimum spin angular momentum along the direction of the magnetic field. A right end resonance involves a transition from the quantum state of maximum spin angular momentum along the direction of the magnetic field. For each quantum state of extreme spin there are two end resonances, a microwave resonance and a Zeeman resonance. The microwave resonance is especially useful for atomic clocks, but it can also be used in magnetometers. The low frequency Zeeman resonance is useful for magnetometers.

CROSS REFERENCE TO RELATED APPLICATION

[0001] This application claims priority to U.S. Provisional ApplicationNo. 60/453,839, filed on Mar. 11, 2003, the disclosure of which ishereby incorporated by reference in its entirety.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The present invention relates to the field of optically pumpedatomic clocks or magnetometers, and more particularly to atomic clocksor magnetomers operating with novel end resonances, which have much lessspin-exchange broadening and much larger signal-to-noise ratios thanthose of conventional resonances.

[0004] 2. Description of the Related Art

[0005] Conventional, gas-cell atomic clocks utilize optically pumpedalkali-metal vapors. Atomic clocks are utilized in various systems whichrequire extremely accurate frequency measurements. For example, atomicclocks are used in GPS (global position system) satellites and othernavigation and positioning systems, as well as in cellular phonesystems, scientific experiments and military applications.

[0006] In one type of atomic clock, a cell containing an active medium,such as rubidium or cesium vapor, is irradiated with both optical andmicrowave power. The cell contains a few droplets of alkali metal and aninert buffer gas at a fraction of an atmosphere of pressure. Light fromthe optical source pumps the atoms of the alkali-metal vapor from aground state to an optically excited state, from which the atoms fallback to the ground state, either by emission of fluorescent light or byquenching collisions with a buffer gas molecule like N₂. The wavelengthand polarization of the light are chosen to ensure that some groundstate sublevels are selectively depopulated, and other sublevels areoverpopulated compared to the normal, nearly uniform distribution ofatoms between the sublevels. It is also possible to excite the sameresonances by modulating the light at the Bohr frequency of theresonance, as first pointed out by Bell and Bloom, W. E. Bell and A. L.Bloom, Phys. Rev. 107, 1559 (1957), hereby incorporated by referenceinto this application. The redistribution of atoms between theground-state sublevels changes the transparency of the vapor so adifferent amount of light passes through the vapor to a photodetectorthat measures the transmission of the pumping beam, or to photodetectorsthat measure fluorescent light scattered out of the beam. If anoscillating magnetic field with a frequency equal to one of the Bohrfrequencies of the atoms is applied to the vapor, the populationimbalances between the ground-state sublevels are eliminated and thetransparency of the vapor returns to its unpumped value. The changes inthe transparency of the vapor are used to lock a clock or magnetometerto the Bohr frequencies of the alkali-metal atoms.

[0007] The Bohr frequency of a gas cell atomic clock is the frequency vwith which the electron spin precesses about the nuclear spin I for analkali-metal atom in its ground state. The precession is caused by themagnetic hyperfine interaction. Approximate clock frequencies arev=6.835 GHz for ⁸⁷Rb and v=9.193 GHz for ¹³³Cs. Conventionally, clockshave used the “0-0” resonance which is the transition between an upperenergy level with azimuthal quantum number 0 and total angular momentumquantum number f=I+½, and a lower energy level, also with azimuthalquantum number 0 but with total angular momentum quantum number f=I−½.

[0008] For atomic clocks, it is important to have the minimumuncertainty, δν, in the resonance frequency ν. The frequency uncertaintyis approximately given by the ratio of the resonance linewidth, Δν, tothe signal to noise ratio, SNR, of the resonance line. That is,δν=ΔνSNR. Clearly, one would like to use resonances with the smallestpossible linewidths, Δν, and the largest possible signal to noise ratio,SNR.

[0009] For miniature atomic clocks it is necessary to increase thedensity of the alkali-metal vapor to compensate for the smaller physicalpath length through the vapor. The increased vapor density leads to morerapid collisions between alkali-metal atoms. These collisions are apotent source of resonance line broadening. While an alkali-metal atomcan collide millions of times with a buffer-gas molecule, like nitrogenor argon, with no perturbation of the resonance, every collision betweenpairs of alkali-metal atoms interrupts the resonance and broadens theresonance linewidth. The collision mechanism is “spin exchange,” theexchange of electron spins between pairs of alkali-metal atoms during acollision. The spin-exchange broadening puts fundamental limits on howsmall such clocks can be. Smaller clocks require larger vapor densitiesto ensure that the pumping light is absorbed in a shorter path length.The higher atomic density leads to larger spin-exchange broadening ofthe resonance lines, and makes the lines less suitable for locking aclock frequency or a magnetometer frequency.

[0010] It is desirable to provide a method and system for reducingspin-exchange broadening in order to make it possible to operate atomicclocks at much higher densities of alkali-metal atoms than conventionalsystems.

SUMMARY OF THE INVENTION

[0011] The present invention relates to a method and system for usingend resonances of highly spin-polarized alkali metal vapors for anatomic clock, magnetometer or other system. A left end resonanceinvolves a transition from the quantum state of minimum spin angularmomentum along the direction of the magnetic field. A right endresonance involves a transition from the quantum state of maximum spinangular momentum along the direction of the magnetic field. For eachquantum state of extreme spin there are two end resonances, a microwaveresonance and a Zeeman resonance. For ⁸⁷Rb, the microwave end resonanceoccurs at a frequency of approximately 6.8 GHz and for ¹³³CS themicrowave end resonance frequency is approximately 9.2 GHz. The Zeemanend resonance frequency is very nearly proportional to the magneticfield. For ⁸⁷Rb the Zeeman end resonance frequency is approximately 700KHz/G, and for ¹³³Cs the Zeeman end resonance frequency is approximately350 KHz/G. The microwave resonance is especially useful for atomicclocks, but it can also be used in magnetometers. The low frequencyZeeman resonance is useful for magnetometers.

[0012] Unlike most spin-relaxation mechanisms, spin-exchange collisionsbetween pairs of alkali metal atoms conserve the total spin angularmomentum (electronic plus nuclear) of the atoms. This causes thespin-exchange broadening of the end resonance lines to approach zero asthe spin polarization P of the vapor approaches its maximum or minimumvalues, P=±1. Spin-exchange collisions efficiently destroy the coherenceof 0-0 transition, which has been universally used in atomic clocks inthe past. As an added benefit, end resonances can have much highersignal-to-noise ratios than the conventional 00 resonance. The highsignal-to-noise ratio occurs because it is possible to optically pumpnearly 100% of the alkali-metal atoms into the sublevels of maximum orminimum angular momentum. In contrast, a very small fraction, typicallybetween 1% and 10% of the atoms, participate in the 00 resonance, sincethere is no simple way to concentrate all of the atoms into either ofthe states between which the 00 resonance occurs. The same high angularmomentum of the quantum states involved in the end resonances accountsfor their relative freedom from resonance line broadening. Spin-exchangecollisions between pairs of alkali-metal atoms, which dominate the linebroadening for the dense alkali-metal vapors needed for miniature,chip-scale atomic clocks, conserve the spin angular momentum. Since thestates for the end transition have the maximum possible angularmomentum, spin-exchange collisions cannot remove the atoms from theirinitial state, because all different final states have lower values ofthe spin angular momentum. None of these advantages accrue to thequantum states of the conventional 0-0 transition.

[0013] The invention will be more fully described by reference to thefollowing drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014]FIG. 1 is a flow diagram of a method of operating an atomic clockin accordance with the teachings of the present invention.

[0015]FIG. 2A is a graph of sublevel populations and susceptibilitiesfor an alkali-metal atom having end clock resonances and nuclear spinI=3/2 with non zero electron spin polarization.

[0016]FIG. 2B is a graph of the relative susceptibilities of FIG. 2A asa function of frequency detuning for each polarization.

[0017]FIG. 3A is a graph of sublevel populations and susceptibilitiesfor an alkali-metal atom having prior art Δm=0 and nuclear spin I=3/2with non zero electron spin polarization.

[0018]FIG. 3B is a graph of the relative susceptibilities of FIG. 3A asa function of frequency detuning for each polarization.

[0019]FIG. 4A is a graph of the amplitude of a prior art resonancesignal for a prior art 0-0 transition of ⁸⁷Rb versus line-widths.

[0020]FIG. 4B is a graph of the prior art amplitude of resonance signalfor a 1-2 transition of ⁸⁷Rb versus line-widths.

[0021]FIG. 5 is a graph of the line-width for the 1-2 hyperfinetransition of ⁸⁷Rb versus an increase in laser power.

[0022]FIG. 6 is a schematic diagram of a system of operating an atomicclock in accordance with the teachings of the present invention.

DETAILED DESCRIPTION

[0023] Reference will now be made in greater detail to a preferredembodiment of the invention, an example of which is illustrated in theaccompanying drawings. Wherever possible, the same reference numeralswill be used throughout the drawings and the description to refer to thesame or like parts.

[0024]FIG. 1 is a flow diagram of a method of operating an atomic clock10 in accordance with the teachings of the present invention. In block12, atoms are generated in an initial state having maximum or minimumspin angular momentum. The quantum numbers f and m are used to label theground-state sublevels of the alkali-metal atom. Here f is the quantumnumber of the total spin, electronic plus nuclear, of the atom, and m,is the azimuthal quantum number, the projection of the total spin alongthe direction of the magnetic field. The possible values of f aref=I+½=a or f=I−½=b, and the possible values of m are m=f, f−1, f−2, . .. , −f For example, for a left end resonance, the initial state i, ofminimum spin angular momentum has the quantum numbers, f_(i),m_(i)=a_(i)−a. For a microwave end resonance, the corresponding finalstate j will have the quantum numbers _(j)f, m_(j)=b, −b. Most of theatoms can be placed in the initial state by pumping the vapor withcircularly polarized light for which the photon spins have one unit ofangular momentum antiparallel to the direction of the magnetic field.The Bohr frequency of the left end resonance is ω−. For example, the endresonance can be a right end resonance. The right end resonancetransition is defined as a transition that occurs between the statesf_(i), m_(i)=a, a and f_(j), m_(j)=b, b with the Bohr frequency of ω+.Most of the atoms can be placed in the initial state by pumping thevapor with circularly polarized light for which the photon spins haveone unit of angular momentum parallel to the direction of the magneticfield.

[0025] In block 14, atoms are generated in a second state having an endresonance by magnetic fields oscillating at the Bohr frequency of atransition from an end state. The magnetic field can oscillate at theBohr frequency ω− or ω+ of the resonance. The atoms can be rubidiumatoms or cesium atoms. The atoms can be pumped with circularlypolarized, D1 resonance light for the rubidium or cesium atoms.Alternatively, in block 16, atoms are generated with end resonances bypumping the atoms with light modulated at the Bohr frequency of atransition from an end state. The light is modulated at the Bohrfrequency ω− or ω+ of the resonance. The atoms can be rubidium atoms orcesium atoms. The atoms can be pumped with modulated, circularlypolarized, D1 resonance light for the Rb or Cs atoms.

[0026] A similar method described above for operating an atomic clockcan be used for operating a magnetometer.

[0027] Hyperfine transitions of the atoms having a first end resonanceand second end resonance are generated by applying radiation at thefirst transition frequency and the second transition frequency. Thefirst transition frequency can be a high frequency resonance that isabout 6.8 GHz for ⁸⁷Rb and 9.2 GHz for ¹³³CS. A similar method can beused for operating a magnetometer in which a low frequency Zeemanresonance is used with a right end resonance and a left end resonance.

[0028] Relaxation due to spin exchange can be analyzed by letting thetime evolution of the spins be due to the combined effects of binaryspin-exchange collisions, as first described by Grossetête, F., 1964, J.Phys. (Paris), 25, 383; 1968, J. Phys. (Paris), 29, 456; Appelt, S. etal., 1998, Phys. Rev. A, 58, 1412 and free evolution in the intervalsbetween collisions. Then the rate of change of the density matrix p isgiven by the non-linear equation, as described in Gibbs, H. M. and Hull,R. J., 1967, Phys. Rev., 153, 132 $\begin{matrix}{{T_{ex}\frac{\rho}{t}} = {{\frac{Tex}{i\quad \hslash}\left\lbrack {H,\rho} \right\rbrack} - {\frac{3}{4}\varphi} + {{S \cdot \rho}\quad S} + {(S) \cdot \left( {\left\{ {S,\rho} \right\} - {2i\quad S \times \rho \quad S}} \right)}}} & (1)\end{matrix}$

[0029] T_(ex) is the mean time between spin-exchange collisions. Foralkali-metal vapors of number density N, the spin exchange rate is1/T_(ex)=κN. The rate coefficient κ≈10⁻⁹ cm⁻³ sec⁻¹ is very nearly thesame for all alkali elements, and has little dependence on temperature,as described in Ressler, N. W., Sands, R. H., and Stark, T. E., 1969,Phys. Rev., 184, 102; Walter, D. K., Griffith, W. M., and Happer, W.,2002, Phys. Rev. Lett., 88, 093004 and Anderson, L. W., Pipkin, F. M.,and Baird, J. C., 1959, Phys. Rev., 116, 87, hereby incorporated byreference into this application.

[0030] The Hamiltonian H of equation (1) is $\begin{matrix}{H = {{{AI} \cdot S} + {g_{S}\mu_{B}{BS}_{z}} - {\frac{\mu_{I}}{I}{{BI}_{z}.}}}} & (2)\end{matrix}$

[0031] A is the coefficient for the magnetic dipole coupling of thenuclear spin I to the electron spin S. For example, in ⁸⁷Rb, A/h=3417.3MHz. The spins are also coupled to an externally applied magnetic fieldof magnitude B, directed along the z-axis of a coordinate system. TheBohr magneton is μ_(B)=9.274×10⁻²⁴ J T⁻¹, and the g value of theelectron is gs=2.0023. The magnetic moment of the alkali nucleus isμ_(r); for example, for the alkali-metal isotope ⁸⁷Rb, μ_(I)=2.75 μ_(N),where μ_(N)=5.051×10⁻²⁷ J T⁻¹ is the nuclear magneton.

[0032] The eigenstates |i> and energies E_(i) of free ground state atomsare defined by the Schrödinger equation

H|i>=E _(i) |i>.   (3)

[0033] The total azimuthal angular momentum operator F_(z)=I_(z)+S_(z)commutes with H, so the eigenstates of equation (3) can be chosen to besimultaneous eigenstates of F_(z),

F _(z) |i>=m _(i) |i>.   (4)

[0034] The azimuthal quantum numbers mi are spaced by unit incrementsbetween the maximum and minimum values, ±(I+½). It is assumed that|B|<<|A/μ_(B)|, so the square of the total angular momentum F=I+S isalso very nearly a good quantum number. Then to good approximation

F·F|i>=f _(i)(f _(i)+1)|i>).   (5)

[0035] The quantum numbers f_(i) can be f_(i)=I+½=a or f_(i)=I−½=b.

[0036] The only non-relaxing solution to equation (1) is thespin-temperature distribution, as described by Anderson, L. W., Pipkin,F. M., and Baird, J. C., 1959, Phys. Rev., 116, 87 and already citedabove. $\begin{matrix}{\rho^{(0)} = {\frac{1}{Z}{^{\beta \quad F_{z}}.}}} & (6)\end{matrix}$

[0037] Substituting equation (6) into equation (1) it is verified that$\begin{matrix}{\frac{\rho^{(0)}}{t} = 0.} & (7)\end{matrix}$

[0038] The spin temperature parameter β is related to the spinpolarization P by $\begin{matrix}{P = {{2{\langle S_{z}\rangle}} = {{\tan \quad h\quad \frac{\beta}{2}\quad {or}\quad \beta} = {\ln {\frac{\quad {1 + P}}{1 - P}.}}}}} & (8)\end{matrix}$

[0039] The partition function of equation (6) is

Z=Tr[e ^(βF) ^(_(z)) ]=Z ₁ Z _(S).   (9)

[0040] For a spin system with spin quantum number J (for example, J=S orJ=I) the partition function is $\begin{matrix}{Z_{J} = {{\sum\limits_{m = {- J}}^{J}\quad ^{\beta \quad m}} = \frac{\sin \quad h\quad {{\beta \lbrack J\rbrack}/2}}{\sin \quad h\quad {\beta/2}}}} & (10) \\{= {\frac{\left( {1 + P} \right)^{J} - \left( {1 - P} \right)^{J}}{2{P\left( {1 - P^{2}} \right)}^{J}}.}} & (11)\end{matrix}$

[0041] Here and subsequently, a spin quantum number in square bracketsdenotes the number of possible azimuthal states, for example, [J]=2J+1.

[0042] The damping is considered of the coherence P_(ij) betweendifferent ground-state sublevels i and j. For example, the coherence canbe induced by radiofrequency magnetic fields, tuned to the Bohrfrequency ω_(ij)=(E_(i)−E_(j))/

between the sublevels |i> and |j>, or by pumping light, modulated at theBohr frequency. Let the polarization state of the atoms be close to thespin temperature limit of equation (6), so that the density matrix canbe written as

ρ=ρ⁽⁰⁾+ρ⁽¹⁾, (2)

[0043] where $\begin{matrix}{\rho^{(1)} = {\sum\limits_{rs}\quad {{r\rangle}\rho_{rs}^{(1)}{{\langle s}.}}}} & (13)\end{matrix}$

[0044] The expectation value of the electronic spin is

<S>=<S> ⁽⁰⁾ +<S> ⁽¹⁾, (14)

[0045] where $\begin{matrix}{{{\langle S\rangle}^{(0)} = {\frac{P}{2}z}},{and}} & (15) \\{{\langle S\rangle}^{(1)} = {\sum\limits_{rs}\quad {{\langle{s{S}r}\rangle}{\rho_{rs}^{(1)}.}}}} & (16)\end{matrix}$

[0046] Substituting equation (12) and equation (14) into equation (1),assuming no time dependence of β or P, and ignoring terms quadraticp⁽¹⁾, it is found that $\begin{matrix}\begin{matrix}{{T_{ex}\frac{\rho^{(1)}}{t}} = {{\frac{T_{ex}}{i\quad \hslash}\left\lbrack {H,\rho^{(1)}} \right\rbrack} - {\frac{3}{4}\rho^{(1)}} + {{S \cdot \rho^{(1)}}S} + {{\langle S\rangle}^{(1)} \cdot \left( {\left\{ {S,\rho^{(0)}} \right\} -} \right.}}} \\{\left. {2i\quad S \times \rho^{(0)}S} \right) + {{\langle S\rangle}^{(0)} \cdot {\left( {\left\{ {S,\rho^{(1)}} \right\} - {2i\quad S \times \rho^{(1)}S}} \right).}}}\end{matrix} & (17)\end{matrix}$

[0047] It is noted that $\begin{matrix}{{{\left\{ {S,\rho^{(0)}} \right\} - {2i\quad S \times \rho^{(0)}S}} = {\frac{^{\beta \quad I_{z}}}{Z}V}},} & (18)\end{matrix}$

[0048] where

V=V(β)={S, e ^(βS) ^(_(z)) }−2iS×e ^(βS) ^(_(z)) S.   (19)

[0049] Accordingly, it can be verified that

V(0)=4S,   (20)

[0050] $\begin{matrix}{{{\frac{V}{\beta}(0)} = 0},} & (21) \\{\frac{^{2}V}{\beta^{2}} = {\frac{1}{4}{V.}}} & (22)\end{matrix}$

[0051] The solution of equation (22) subject to the boundary conditionsof equation (20) and equation (21) is $\begin{matrix}{V = {4S\quad \cosh {\frac{\beta}{2}.}}} & (23)\end{matrix}$

[0052] Substituting equation (18) and equation (23) into equation (17)and taking the matrix element between the states i and j, it is foundthat $\begin{matrix}{\frac{\rho_{ij}^{(1)}}{t} = {{{- \left( {\gamma_{ij} + {\quad \omega_{ij}}} \right)}\rho_{ij}^{(1)}} - {\sum\limits_{{rs} \neq {ij}}^{\quad}\quad {\Gamma_{{ij};{rs}}{\rho_{rs}^{(1)}.}}}}} & (24)\end{matrix}$

[0053] The damping rate γ_(ij)=Γ_(ij;ij), is given by $\begin{matrix}\begin{matrix}{{T_{ex}\gamma_{ij}} = {\frac{3}{4} - {{\langle{i{S_{z}}i}\rangle}{\langle{j{S_{z}}j}\rangle}} - {\frac{2}{Z_{I}}{{\langle{j{S}i}\rangle} \cdot {\langle{i{{S\quad ^{\beta \quad I_{z}}}}j}\rangle}}} -}} \\{{\frac{P}{2}{\left( {{\langle{i{S_{z}}i}\rangle} + {\langle{j{S_{z}}j}\rangle}} \right).}}}\end{matrix} & (25)\end{matrix}$

[0054] For the low-field limit, the projection theorem for coupledangular momenta can be used, as described in Appelt, S., Ben-AmarBaranga, Young, A. R., and Happer, W., 1999, Phys. Rev. A, 59, 2078 toevaluate matrix elements of S_(z), and it is found that $\begin{matrix}{{{{\langle{i{S_{z}}i}\rangle}{\langle{j{S_{z}}j}\rangle}} = \frac{{4{\overset{\_}{m}}^{2}} - 1}{{4\lbrack I\rbrack}^{2}}},} & (26) \\{and} & \quad \\\begin{matrix}{{\frac{2}{Z_{I}}{{\langle{j{S}i}\rangle} \cdot {\langle{i{{S\quad ^{\beta \quad I_{z}}}}j}\rangle}}} = {\frac{1}{Z_{I}}{{\langle{j{S_{-}}i}\rangle} \cdot {\langle{i{{S_{+}\quad ^{\beta \quad I_{z}}}}j}\rangle}}}} \\{= {Q_{\overset{\_}{m}}{{\langle{j{S_{-}}i}\rangle} \cdot {{\langle{i{S_{+}\quad }j}\rangle}.}}}}\end{matrix} & (27) \\{Here} & \quad \\{Q_{\overset{\_}{m}} = {{Q_{\overset{\_}{m}}(P)} = {\frac{^{\beta \quad \overset{\_}{m}}}{Z_{I}} = \frac{2{P\left( {1 + P} \right)}^{I + \overset{\_}{m}}\left( {1 - P} \right)^{I - \overset{\_}{m}}}{\left( {1 + P} \right)^{I} - \left( {1 - P} \right)^{I}}}}} & (28)\end{matrix}$

[0055] is the probability, as described in Appelt, S., Ben-Amar Baranga,Young, A. R., and Happer, W., 1999, Phys. Rev. A, 59, to find thenucleus with the azimuthal number {overscore (m)} for thespin-temperature distribution of equation (6). The following symmetry isnoted

Q{overscore (m)}(P)=Q−{overscore (m)}(−P).   (29)

[0056] Using the projection theorem, as described in Varshalovich, D.A., Moskalev, A. N., and Khersonskii, V. K., 1988, Quantum Theory ofAngular Momentum (Singapore:World Sci.), hereby incorporated byreference into this application, it is found that $\begin{matrix}{{{{\langle{j{S_{-}}i}\rangle}{\langle{i{S_{+}}j}\rangle}} = \frac{\lbrack f\rbrack^{2} - {4{\overset{\_}{m}}^{2}}}{{4\lbrack I\rbrack}^{2}}},} & (30) \\\begin{matrix}{{and}\quad {also}} \\{{{\langle{i{S_{Z}}i}\rangle} + {\langle{j{S_{Z}}j}\rangle}} = {\frac{2\overset{\_}{m}}{\lbrack I\rbrack}{\left( {- 1} \right)^{a - f}.}}}\end{matrix} & (31)\end{matrix}$

[0057] The damping rate is therefore $\begin{matrix}\begin{matrix}{{T_{ex}\gamma_{f,{{\overset{\_}{m} + {1/2}};f},{\overset{\_}{m} - {1/2}}}} = {\frac{3}{4} - \frac{{4{\overset{\_}{m}}^{2}} - 1}{{4\lbrack I\rbrack}^{2}} - {Q_{\overset{\_}{m}}\frac{\lbrack f\rbrack^{2} - {4{\overset{\_}{m}}^{2}}}{{4\lbrack I\rbrack}^{2}}} -}} \\{{\frac{P\quad \overset{\_}{m}}{\lbrack I\rbrack}{\left( {- 1} \right)^{a - f}.}}}\end{matrix} & (32)\end{matrix}$

[0058] Equation (32) predicts that the spin-exchange damping rate of theZeeman “end” transition with f=a and {overscore (m)}=I vanishes as P→1.

[0059] The damping of resonances with f_(i)=a=I+½, m_(i)={overscore(m)}+½ and f_(i)=b=I−½, m_(j)={overscore (m)}−½, are excited by magneticfields, oscillating at right angles to the z axis, and at frequencies onthe order of the hyperfine frequency ^(v)hf=[I]A/2h, (^(v)hf=6834.7 MHzfor ⁸⁷Rb) or by pumping light modulated at the same frequency. As in theease of low-field Zeeman resonances described above, it is found that$\begin{matrix}{{{\langle{i{S_{Z}}i}\rangle}{\langle{j{S_{Z}}j}\rangle}} = {\frac{{4{\overset{\_}{m}}^{2}} - 1}{{4\lbrack I\rbrack}^{2}}.}} & (33)\end{matrix}$

[0060] Equation (27) remains valid for the high-field Zeeman resonances.Using the Wigner-Eckart theorem in the form given by Varshalovich, it isfound that $\begin{matrix}{{\langle{i{S_{+}}j}\rangle} = {{- \sqrt{\frac{2}{\lbrack a\rbrack}}}{\langle{a{S}b}\rangle}{C_{{bm}_{j}:11}^{{am}_{i}}.}}} & (34)\end{matrix}$

[0061] The reduced matrix element is described in Varshalovich, and is$\begin{matrix}{{\langle{a{S}b}\rangle} = {- {\sqrt{\frac{\lbrack I\rbrack^{2} - 1}{2\lbrack I\rbrack}}.}}} & (35)\end{matrix}$

[0062] The Clebsh-Gordon coefficient is given by Table 8.2 ofVarshalovich and is $\begin{matrix}{C_{{bm}_{j}:11}^{{am}_{i}} = {\sqrt{\frac{\left( {\lbrack I\rbrack + {2\overset{\_}{m}}} \right)^{2} - 1}{{4\lbrack I\rbrack}\left( {\lbrack I\rbrack - 1} \right)}}.}} & (36)\end{matrix}$

[0063] In analogy to equation (31), it is found $\begin{matrix}{{{\langle{i{S_{Z}}i}\rangle} + {\langle{j{S_{Z}}j}\rangle}} = {\frac{1}{\lbrack I\rbrack}.}} & (37)\end{matrix}$

[0064] The damping rate is therefore $\begin{matrix}\begin{matrix}{{T_{ex}\gamma_{a,{{\overset{\_}{m} + {1/2}};b},{\overset{\_}{m} - {1/2}}}} = {\frac{3}{4} + \frac{{4{\overset{\_}{m}}^{2}} - 1}{{4\lbrack I\rbrack}^{2}} -}} \\{{{Q_{\overset{\_}{m}}\frac{\left( {\lbrack I\rbrack + {2\overset{\_}{m}}} \right)^{2} - 1}{{4\lbrack I\rbrack}^{2}}} - {\frac{P}{2\lbrack I\rbrack}.}}}\end{matrix} & (38)\end{matrix}$

[0065] The damping rates for the resonances with f_(i)=a=I+½,m_(i)={overscore (m)}−{fraction (1/2 )} f_(i)=b=I −½, m_(j)={overscore(m)}+½, can be similarly calculated, and it is found $\begin{matrix}\begin{matrix}{{T_{ex}\gamma_{a,{{\overset{\_}{m} - {1/2}};b},{\overset{\_}{m} + {1/2}}}} = {\frac{3}{4} + \frac{{4{\overset{\_}{m}}^{2}} - 1}{{4\lbrack I\rbrack}^{2}} -}} \\{{{Q_{\overset{\_}{m}}\frac{\left( {\lbrack I\rbrack + {2\overset{\_}{m}}} \right)^{2} - 1}{{4\lbrack I\rbrack}^{2}}} + {\frac{P}{2\lbrack I\rbrack}.}}}\end{matrix} & (39)\end{matrix}$

[0066] For conventional atomic clocks, unpolarized pumping light with anappropriate frequency profile generates hyperfine polarization (I·S),and the clock is locked to the frequency of the “field-independent 0-0”transition between the states f_(i), m_(i)=a, 0 and f_(j), m_(j)=b, 0.The density matrix for the alkali-metal atoms in a conventional atomicclock can therefore be described by equation (12) but with ) ρ⁽⁰⁾ givenby $\begin{matrix}{\rho^{(0)} = {\frac{1}{2\lbrack I\rbrack} + {\frac{2{\langle{I \cdot S}\rangle}{I \cdot S}}{{I\left( {I + 1} \right)}\left( {{2I} + 1} \right)}.}}} & (40)\end{matrix}$

[0067] Unlike the spin-temperature distribution of equation (6), whichis unaffected by spin-exchange collisions, the hyperfine polarization ofequation (40) relaxes at the spin exchange rate; that is, if equation(40) is substituted into equation (1) it is found that $\begin{matrix}{\frac{{\langle{I \cdot S}\rangle}}{t} = {{- \frac{1}{T_{ex}}}{{\langle{I \cdot S}\rangle}.}}} & (41)\end{matrix}$

[0068] Substituting equation (12) with equation (40) into equation (1),and writing only the self-coupling term explicitly, it is found that inanalogy to equation (24) $\begin{matrix}{\frac{\rho_{{am}:{bm}}^{(1)}}{t} = {{- \left( {\gamma_{{am}:{bm}} + {\quad \omega_{{am}:{bm}}}} \right)}\rho_{{am}:{bm}^{-}}^{(1)}{\ldots \quad.}}} & (42)\end{matrix}$

[0069] The damping rate γ_(am); _(bm) is independent of the hyperfinepolarization(I·S), and is given by $\begin{matrix}\begin{matrix}{{T_{ex}\gamma_{{am}:{bm}}} = {\frac{3}{4} - {{\langle{{am}{S_{z}}{am}}\rangle}{\langle{{bm}{S_{z}}{bm}}\rangle}} -}} \\{{{\frac{2}{\lbrack I\rbrack}{\langle{{bm}{S_{z}}{am}}\rangle}{\langle{{am}{S_{z}}{bm}}\rangle}},}}\end{matrix} & (43)\end{matrix}$

[0070] which yields $\begin{matrix}{\gamma_{{am}:{bm}} = {{\frac{1}{T_{ex}}\left\lbrack {\frac{{3\lbrack I\rbrack} - 2}{4\lbrack I\rbrack} + {\left( {\lbrack I\rbrack + 2} \right)\frac{m^{2}}{\lbrack I\rbrack^{3}}}} \right\rbrack}.}} & (44)\end{matrix}$

[0071] Resonance frequencies ω_(ij)=(E_(i)−E_(j))/

of clock, transitions can be determined. The ground-state energiesE_(fm) of the Hamiltonian equation (2) are $\begin{matrix}{{E_{fm} = {{- \frac{\hslash \quad \omega_{hf}}{2\lbrack I\rbrack}} - {{\frac{\mu_{1}}{I}{Bm}} \pm {\frac{\hslash \quad \omega_{hf}}{2}\sqrt{1 + {\frac{4m}{\lbrack I\rbrack}x} + x^{2}}}}}},} & (45)\end{matrix}$

[0072] where ω_(hf)=A[I]/2

is the zero-field ground-state hyper-fine frequency and $\begin{matrix}{x = \frac{\left( {{g_{S}µ_{B}} + {µ_{I}/I}} \right)B}{{\hslash \omega}_{hf}}} & (46)\end{matrix}$

[0073] is the Breit-Rabi parameter. The ±signs of equation (45)correspond to the sublevels with f=a=I+½ and f=b=I −½, respectively. Theresonance frequencies ω_(am;bm), correct to second order in the magneticfield, are given by $\begin{matrix}{\omega_{{am}:{bm}} = {{\omega_{hf}\left\lbrack {1 + {\frac{2m}{\lbrack I\rbrack}x} + {\frac{1}{2}\left( {1 - \frac{4m^{2}}{\lbrack I\rbrack^{2}}} \right)x^{2}}} \right\rbrack}.}} & (47)\end{matrix}$

[0074] The frequency ω_(a0; b0) of the 0-0 clock transition depends onthe magnetic field only in second order and is $\begin{matrix}{\omega_{{a0}:{b0}} = {{\omega_{hf}\left( {1 + \frac{x^{2}}{2}} \right)}.}} & (48)\end{matrix}$

[0075] According to equation (44), the damping rate of this transitionis $\begin{matrix}{\gamma_{{a0}:{b0}} = {\frac{1}{T_{ex}}{\left( {\frac{3}{4} - \frac{1}{2\lbrack I\rbrack}} \right).}}} & (49)\end{matrix}$

[0076] The resonance frequencies ^(ω)a, m±1:2: b, m ∓½ of transitionsbetween states (a, {overscore (m)}±½) and (b, {overscore (m)}±½)are$\begin{matrix}\begin{matrix}{\omega_{a,{{m \pm {1/2}}:b},{m \mp {1/2}}} = {{\mp \frac{µ_{I}B}{I\quad \hslash}} +}} \\{{{\omega_{hf}\left\lbrack {1 + {\frac{2\overset{\_}{m}}{\lbrack I\rbrack}x} + {\frac{1}{2}\left( {1 - \frac{{4{\overset{\_}{m}}^{2}} + 1}{\lbrack I\rbrack^{2}}} \right)x^{2}}} \right\rbrack}.}}\end{matrix} & (50)\end{matrix}$

[0077] The frequencies ω+ and ω− of the “end” transitions (a, a)

(b, b), with {overscore (m)}=I, and (a, −a)

(b, −b), with {overscore (m)}=−I, are given by $\begin{matrix}{{\omega_{+} = {{\omega_{hf}\left( {1 + {\frac{2I}{\lbrack I\rbrack}x} + {\frac{2I}{\lbrack I\rbrack^{2}}x^{2}}} \right)} - \frac{µ_{I}B}{I\quad \hslash}}},{\omega_{-} = {{\omega_{hf}\left( {1 - {\frac{2I}{\lbrack I\rbrack}x} + {\frac{2I}{\lbrack I\rbrack^{2}}x^{2}}} \right)} + {\frac{µ_{I}B}{I\quad \hslash}.}}}} & (51)\end{matrix}$

[0078] While the frequencies of the “end” transitions are linear in themagnetic field, their average, $\begin{matrix}{\overset{\_}{\omega} = {{\omega_{hf}\left( {1 + {\frac{2I}{\lbrack I\rbrack^{2}}x^{2}}} \right)} + {O\left( x^{4} \right)}}} & (52)\end{matrix}$

[0079] is field-independent to first-order, similar to the frequency ofthe conventional 0-0 transition, and the term quadratic in x is a factor4I/[I]² smaller compared to the corresponding term for the 0-0transition. The difference between the frequencies of the “end”transitions, $\begin{matrix}{{\Delta \omega} = {{\omega_{+} - \omega_{-}} = {{\frac{2\left( {{2I\quad g_{S}µ_{B}} - {µ_{I}/I}} \right)}{\lbrack I\rbrack \hslash}B} + {O\left( B^{3} \right)}}}} & (53)\end{matrix}$

[0080] is proportional to the external magnetic field and can be used tomeasure or lock the field.

[0081] A small magnetic field of amplitude B₁, oscillating at thefrequency ω≈ω_(ij) and polarized for maximum coupling of the states |i>and |j>, will induce an oscillating electronic spin with the samepolarization and with an amplitude <S>₁∝χ_(ij)B₁. The relativesusceptibility is $\begin{matrix}{X_{ij} = {\frac{\left( {\rho_{ii}^{(0)} - \rho_{jj}^{(0)}} \right){{\langle{i{S}j}\rangle}}^{2}}{T_{ex}\left\lbrack {\left( {\omega_{ij} - \omega} \right) - {i\gamma}_{ij}} \right\rbrack}.}} & (54)\end{matrix}$

[0082]FIG. 2A illustrates sublevel populations and susceptibilities foran alkali-metal atom of nuclear spin I=3/2 with non-zero electron spinpolarization. The two bar-graph diagrams show, for electron spinpolarizations P=2<Sz>=0.1 and P=0.8, the populations of the magneticsub-levels |i> of the Hamiltonian equation (2) for the spin-temperaturedistribution p⁽⁰⁾ of equation (6). The upper five levels have totalangular momentum quantum number f=2; the lower three, f=1. Beneath eachlevel the corresponding azimuthal quantum number m is given. FIG. 2Bbelow FIG. 2A shows the imaginary parts X_(ij) of the relativesusceptibilities as a function of the frequency detuning ω−ω_(hf) foreach polarization P. The susceptibility at polarization P=0.1 has beenmagnified one-hundredfold. The transition to which each resonancecorresponds is indicated in the population diagrams by the arrowsdirectly above the resonances. It has been found that as thepolarization increases from 0.1 to 0.8, the amplitude of the aa

bb resonance increases by a factor of 122, while its width decrease by afactor of 5.7.

[0083] Accordingly, the end resonance, that is the resonance for thecoupled states |i>=|aa> and |j>=|bb>, depends strongly on thepolarization of the vapor. The amplitude of the transition increases bya factor 122 as the spin polarization P=2<Sz> increases from 0.1 to 0.8.This is because the population difference ρ_(ii)−ρ_(jj) approaches itsmaximum value of 1 as P→1.

[0084] From FIG. 2A, it is shown that there is a substantial narrowingof the end resonance for high polarizations and spin-exchange broadeningvanishes in the limit that P→1. The narrowing occurs becausespin-exchange collisions conserve spin angular momentum. The upper stateof the end resonance has maximum spin angular momentum. Spin-exchangecollisions must couple a (two-atom) initial state to a final state ofthe same spin angular momentum. In the high polarization limit, thereare no final states with the same angular momentum as the initial state,and the scattering is suppressed.

[0085] The spin-temperature distributions ρ⁽⁰⁾ shown as bar graphs onthe top of FIG. 2A and given by equation (6), do not relax at all underthe influence of spin-exchange collisions. The spin-temperaturedistributions of FIG. 2A can therefore be maintained with relativelyweak, circularly polarized pumping light which is just enough tocompensate for diffusion of the atoms to the cell walls ordepolarization due to collisions with buffer-gas atoms such as nitrogen,helium and the like. The weak light causes little light-broadening ofthe resonance line-widths.

[0086]FIG. 3A illustrates the amplitudes of a prior art clock resonancewith Δm=0. Sublevel populations and susceptibilities for an alkali-metalatom of nuclear spin I=3/2 with non-zero hyperfine polarization. The twobar-graph diagrams show, for hyperfine polarizationsP_(HF)=−2<I·S>/(I+1)=0.1 and P_(HF)=0.8, the populations of the magneticsub-levels |i> of the Hamiltonian equation (2) for thehyperfine-polarized distribution ρ⁽⁰⁾ of equation (40). FIG. 3B belowFIG. 3A shows the imaginary parts of the relative susceptibilities as afunction of the frequency detuning ω−ω_(hf) for each of these twohyperfine polarizations. The transition to which each resonancecorresponds is indicated in the population diagrams by the arrowsdirectly above the resonances. As the hyperfine polarization increasesfrom 0.1 to 0.8, the amplitude of each am ++bm resonance increases by afactor of 8, but their widths remain constant.

[0087] Accordingly, the amplitude of the 0-0 clock resonance increasesby a factor of 8 when the magnitude of the hyperfine polarization <I·S>increases by a factor of 8. This is a much smaller increase than for theend transition shown in FIG. 2B, where the amplitude increases by afactor of 122 for an eightfold increase in polarization. The value ofthe population differences ρ_(ii) ⁽⁰⁾−ρ_(jj) ⁽⁰⁾ for the 0-0conventional clock transition can not exceed the inverse of themultiplicity [a]=[I]+1 of the upper Zeeman multiplet or [b]=[I]−1 of thelower Zeeman multiplet.

[0088] From FIGS. 3A-3B or equation (44), we see that the conventional,0-0 clock resonance has a line-width that is independent of thehyperfine polarization −I·S>. The line-width is a large fraction,(3[I]−2)/(4[I]) of the spin-exchange rate.

[0089] The population distributions ρ⁽⁰⁾ of equation (40) shown as bargraphs in FIG. 3A, relax at the spin-exchange rate 1/T_(ex). This ratecan be much faster than the diffusion rate to the walls or thedepolarization rates due to collisions with buffer-gas atoms. Tomaintain a substantial hyperfine polarization, the optical pumping ratemust be comparable to or greater than the spin-exchange rate. Theresonance lines will therefore have substantial light broadening, and itwill be increasingly difficult to maintain high polarizations at thehigh vapor densities needed for very small atomic clocks.

[0090] To produce a substantial hyperfine polarization <I·S> which isneeded for the conventional 0-0 clock transition, the pressurebroadening of the absorption lines must not exceed the hyperfinesplitting of the optical absorption lines, since the pumping depends ondifferential absorption from the ground state multiplets of total spinangular momentum a and b. Accordingly, high buffer-gas pressuresseriously degrade the optical pumping efficiency for conventionalclocks. In contrast, high buffer-gas pressures do not degrade theoptical pumping efficiency of an atomic clock, based on left and rightend transitions of the present invention which can be generated bypumping with left-and right-circularly polarized D₁ light.

[0091]FIGS. 4A-4B illustrate the resonance signal for a respective 0-0transition and an end 1-2 transition of ⁸⁷Rb. FIG. 4A illustrates thatthe resonance signal for a prior art 0-0 hyperfine transition of ⁸⁷Rbhas line broadening. FIG. 4B illustrates that the resonance signal forthe 1-2 hyperfine end transition of ⁸⁷Rb has light narrowing and anincreased signal to noise ratio. The results were determined from cellT108 (N₂700 torr+⁸⁷Rb) at 140° C., thickness 2 mm, laser frequency377104 GHz, beam diameter 6 mm, microwave fc=6.8253815 GHz −5 dbm, Bfield˜4.6 Gauss, 1-2 transition.

[0092]FIG. 5 illustrates a graph of the line-width for the 1-2 hyperfinetransition of ⁸⁷Rb versus an increase in laser power. It is shown thathigher intensity of circularly polarized laser light polarizes Rb vaporinside the cell and narrows the line-width of hyperfine end transitionsfor the 1-2 hyperfine transition.

[0093]FIG. 6 is a schematic diagram of a system for operating an atomicclock 30. Vapor cell 32 contains atoms of material which have ahyperfine resonance transition that occurs between a left end resonance.Vapor cell 34 contains atoms of a material which have a hyperfineresonance transition that occurs between a right end resonance. Suitablematerials include cesium or rubidium. Conventional means can be usedwith cell 32 and cell 34 for stabilizing the magnetic B field andtemperature. Vapor cells 32 and 34 can include buffer gases inside thecells to suppress frequency shift due to temperature drift.

[0094] Laser diode 35 generates a beam of left circularly polarizedlight which pumps atomic vapor in cell 32 to maximize the left endresonance. Laser diode 36 generates a beam of right circularly polarizedlight which pumps atomic vapor in cell 34 to maximize the right endresonance.

[0095] Control signal designed to cause a change in state of the atomsin cell 32 is applied as to input 37. Control signal designed to cause achange in state of the atoms in cell 34 is applied as input 38. Controlsignals can be generated by a frequency oscillator and hyperfineresonance lock loop. Photo detectors 39 and 40 detect radiation fromrespective cells 32 and 34.

[0096] A similar system described above for operating an atomic clockcan be used for operating a magnetometer.

[0097] It is to be understood that the above-described embodiments areillustrative of only a few of the many possible specific embodimentswhich can represent applications of the principles of the invention.Numerous and varied other arrangements can be readily devised inaccordance with these principles by those skilled in the art withoutdeparting from the spirit and scope of the invention.

What is claimed is:
 1. A method for operating an atomic clock comprising the steps of: generating atoms in a ground-state sublevel of maximum or minimum spin from which end resonances can be excited; and exciting magnetic resonance transitions in the atoms with magnetic fields oscillating at Bohr frequencies of the end resonances.
 2. The method of claim 1 wherein the magnetic field oscillates at the Bohr frequency ω− of the resonance.
 3. The method of claim 1 wherein the magnetic field oscillates at the Bohr frequency ω+ of the resonance.
 4. The method of claim 1 wherein said atoms are rubidium atoms or cesium atoms.
 5. The method of claim 4 wherein the atoms are pumped with circularly polarized, D1 resonance light for the rubidium or cesium atoms.
 6. A method for operating an atomic clock comprising the steps of: generating atoms in a ground-state sublevel of maximum or minimum spin; and pumping the atoms with light modulated at a Bohr frequency of the end resonance for exciting transitions in the atoms.
 7. The method of claim 6 wherein the light is modulated at the Bohr frequency ω− of the resonance.
 8. The method of claim 6 wherein the light is modulated at the Bohr frequency ω+ of the resonance.
 9. The method of claim 6 wherein said atoms are rubidium atoms or cesium atoms.
 10. The method of claim 6 wherein the atoms are pumped with modulated, circularly polarized, D1 resonance light for the Rb or Cs atoms.
 11. A system for operating an atomic clock comprising: means for generating atoms in a ground-state sublevel of maximum or minimum spin from which end resonances can be excited; and means for generating hyperfine transitions of said atoms by applying magnetic fields oscillating at Bohr frequencies of the end resonances.
 12. The system of claim 11 wherein the magnetic field oscillates at the Bohr frequency ω− of the resonance.
 13. The system of claim 11 wherein the magnetic field oscillates at the Bohr frequency ω+ of the resonance.
 14. The system of claim 11 wherein said atoms are rubidium atoms or cesium atoms.
 15. The system of claim 14 wherein the atoms are pumped with circularly polarized, D1 resonance light for the rubidium or cesium atoms.
 16. A system for operating an atomic clock comprising: means for generating atoms in a ground-state sublevel of maximum or minimum spin, from which end resonances can be excited; and means for pumping the atoms with light modulated at a Bohr frequency of the end resonance for exciting transitions in the atoms.
 17. The system of claim 16 wherein the light is modulated at the Bohr frequency ω− of the resonance.
 18. The system of claim 16 wherein the light is modulated at the Bohr frequency ω+ of the resonance.
 19. The system of claim 12 wherein said atoms are rubidium atoms or cesium atoms.
 20. The system of claim 19 wherein the atoms are pumped with modulated, circularly polarized, D1 resonance light for the rubidium or cesium atoms.
 21. A method for operating a magnetometer comprising the steps of: generating atoms in a ground-state sublevel of maximum or minimum spin from which end resonances can be excited; and exciting magnetic resonance transitions in the atoms with magnetic fields oscillating at Bohr frequencies of the end resonances.
 22. The method of claim 21 wherein the magnetic field oscillates at the Bohr frequency ω− of the resonance.
 23. The method of claim 21 wherein the magnetic field oscillates at the Bohr frequency ω+ of the resonance.
 24. The method of claim 21 wherein said atoms are rubidium atoms or cesium atoms.
 25. The method of claim 24 wherein the atoms are pumped with circularly polarized, D1 resonance light for the rubidium or cesium atoms.
 26. A method for operating a magnetometer comprising the steps of: generating atoms in a ground-state sublevel of maximum or minimum spin; and pumping the atoms with light modulated at a Bohr frequency of the end resonance for exciting transitions in the atoms.
 27. The method of claim 26 wherein the light is modulated at the Bohr frequency ω− of the resonance.
 28. The method of claim 26 wherein the light is modulated at the Bohr frequency ω+ of the resonance.
 29. The method of claim 26 wherein said atoms are rubidium atoms or cesium atoms.
 30. The method of claim 29 wherein the atoms are pumped with modulated, circularly polarized, D1 resonance light for the rubidium or cesium atoms.
 31. A system for operating a magnetometer comprising: means for generating atoms in a ground-state sublevel of maximum or minimum spin from which end resonances can be excited; and means for generating hyperfine transitions of said atoms by applying magnetic fields oscillating at Bohr frequencies of the end resonances.
 32. The system of claim 31 wherein the magnetic field oscillates at the Bohr frequency ω− of the resonance.
 33. The system of claim 31 wherein the magnetic field oscillates at the Bohr frequency ω+ of the resonance.
 34. The system of claim 31 wherein said atoms are rubidium atoms or cesium atoms.
 35. The system of claim 31 wherein the atoms are pumped with circularly polarized, D1 resonance light for the rubidium or cesium atoms.
 36. A system for operating a magnetometer comprising: means for generating atoms in a ground-state sublevel of maximum or minimum spin, from which end resonances can be excited; and means for pumping the atoms with light modulated at a Bohr frequency of the end resonance for exciting transitions in the atoms.
 37. The system of claim 36 wherein the light is modulated at the Bohr frequency ω− of the resonance.
 38. The system of claim 36 wherein the light is modulated at the Bohr frequency ω+ of the resonance.
 39. The system of claim 36 wherein said atoms are rubidium atoms or cesium atoms.
 40. The system of claim 36 wherein the atoms are pumped with modulated, circularly polarized, D1 resonance light for the rubidium or cesium atoms. 